While the invention may be applied to a range of applications, consider as an example the manufacture of integrated circuits (ICs) by a lithographic process. In that instance, a lithographic apparatus is used to apply a pattern of device features to be formed on an individual layer of the IC. This pattern can be transferred onto a target portion (e.g., including part of, one, or several dies) on a substrate (e.g., a silicon wafer).
In lithographic processes, it is desirable frequently to make measurements of the structures created, e.g., for process control and verification. Various tools for making such measurements are known, including scanning electron microscopes (SEM), which are often used to measure critical dimension (CD). Other specialized tools are used to measure parameters related to asymmetry. One of these parameters is overlay, the accuracy of alignment of two layers in a device. Recently, various forms of scatterometers have been developed for use in the lithographic field. These devices direct a beam of radiation onto a target and measure one or more properties of the scattered radiation as it is reflected and/or transmitted by the target, e.g., intensity at a single angle as a function of wavelength; intensity at one or more wavelengths as a function of angle; or polarization as a function of angle—to obtain a “spectrum” of one form or another. The term “spectrum” in this context will be used with a wide scope. It may refer to a spectrum of different wavelengths (colors), it may refer to a spectrum of different directions (diffraction angles), different polarizations, or a combination of any or all of these. From this spectrum a property of interest of the target can be determined. Determination of the property of interest may be performed by various techniques. One particular approach is to perform reconstruction of the target structure by iterative calculations. A mathematical model of the target is created and calculations are performed to simulate interaction of radiation with the target. Parameters of the model are adjusted and calculations repeated until the simulated spectrum becomes the same as the observed spectrum. The adjusted parameter values then serve as a measurement of the real target structure. Each updated model represents a point in “parameter space”, which is a mathematical space with as many dimensions as there are parameters in the model. The aim of the iterative process is to converge to a point in parameter space that represents, at least approximately, the parameters of the actual target structure. In another approach, simulated spectra are calculated in advance for a variety of points in the parameter space. These simulated spectra serve as a “library” which is searched to find a match for a spectrum observed later on a real target.
Compared with SEM techniques, scatterometers can be used with much higher throughput, on a large proportion or even all of the product units. The measurements can be performed very quickly. On the other hand, reconstruction requires a great deal of computation. For example, in many applications, the FMM method in one-dimensional models requires a calculation of order N^3 (N cubed) operations, for a given number N of harmonics. (In two-dimensional models the calculation is of order Nx^3*Ny^3.) As the sizes of features produced by lithography shrink ever smaller and dimensional tolerances shrink accordingly, there is an interest in the use of diffraction based techniques (scatterometry) at shorter wavelengths, such as x-ray and “soft x-ray” (extreme ultraviolet) wavelengths. Scattering of electromagnetic waves can be simulated by use of Maxwell's equations at such short wavelengths in the same way as at longer wavelengths. This approach is used to analyze x-ray diffraction patterns from all field of physics: power diffraction, crystallography, biology, etc. In semiconductor manufacturing, reconstruction of critical dimension using small-angle x-ray scattering (CD-SAXS) is already known. Examples are in references (1) (Lemaillet 2013) and (2)(Jones 2003).
Examples of transmissive and reflective metrology techniques using these wavelengths in transmissive and/or reflective scattering modes are disclosed in pending patent applications PCT/EP2015/058238 filed 16 Apr. 2015, EP15180807.8 filed 12 Aug. 2015 and EP15180740.1 filed 12 Aug. 2015, not published at the present priority date.
In the x-ray and EUV range of the electromagnetic spectrum, many materials become quite transparent. In other words, the contrast between the materials in the patterned layer is small. (The patterned may be made of two or more solid materials, or there may be a pattern in one material in an atmosphere, such as air, vacuum or He, as is known in the x-ray and EUV fields. In this regime, a technique known as the Born approximation has been used in the References (3) (Li 2010) and (4) (Sinha 1988) to simplify the model and its numerical solution. Non-patent references are listed in full at the end of this description. In this case a scattered radiation term in the Maxwell equations, which is defined as the product of the contrast permittivity and the total field, is approximated by the product of the contrast permittivity and the background (or incident) field. In the references, the Born approximation is typically implemented through integral methods (see for example References (5) (Van den Berg 1984) and (6) (Trattner 2009).
In order to find the numerical solution of the integral formulation of the Born approximation, a full discretization (in all spatial dimensions x, y, z) is required. In typical applications, the domain size (that is, the spatial period of the model structure, or other spatial extent) is much larger than the wavelength. For example, the spatial period of a target grating might be ten or so nanometers while the radiation wavelength is a fraction of a nanometer. If we denote by sx/sy/sz the ratio between domain size in the x/y/z direction and the wavelength of radiation used, then the number of degrees of freedom (the number of unknowns in the discretized Maxwell's equations) is proportional to sx*sy*sz. Since this product can easily reach a factor of 1000, the computation might become expensive. Moreover, due to discretization in the z direction, the solution in the known integral methods is prone to approximation errors in all directions.
As an alternative to integral methods for simulating interaction of radiation with different structures, modal methods are also known, where the radiation field and the structure are transformed into a mode space. An example of a modal method is the Fourier modal method (FMM) or rigorous coupled wave analysis or RCWA. RCWA is well-known and suitable for application to periodic structures, especially. Modes in the Fourier modal method are also referred to as “harmonics”. Techniques for extending the application of FMM to aperiodic structures (including “finite periodic” structures), are described in Reference (7) M Pisarenco and others, “Aperiodic Fourier modal method in contrast-field formulation for simulation of scattering from finite structures”, J. Opt. Soc. Am. A, Vol. 27, No. 11 (November 2010), pp 2423-2431. A further paper on this topic by the same authors is (8) “Modified S-matrix algorithm for the aperiodic Fourier modal method in contrast-field formulation”, J. Opt. Soc. Am. A 28, 1364-1371 (2011). The contents of both papers are hereby incorporated by reference. According to these references, complexity in the calculations is reduced by considering separately a background permittivity and a contrast permittivity, and calculating a background radiation field and a scattered radiation field. Representing each of these fields in Fourier space allows an analytical solution of Maxwell's equations in one dimension.